I've been listening to the JFK tower feed, and the tower controller will always issue "taxi to the gate on Alpha monitor ground .65" or whatever to aircraft that have just landed. Does this mean to stay with the tower and....listen to ground, switch over to ground completely, or what?

B752, this means to switch your radio to 121.65, and listen on that frequency for any transmissions to you but it is not requested to CONTACT ground to acknowledge that you're monitoring. Simply reading back the "ground .65" transmission to tower is sufficient. One monitors ground in case taxi instructions need to be amended, but since you've already gotten taxi instructions from tower it's not needed to make a transmission on the ground frequency. Also, by telling you to monitor ground, you're given permission to no longer be available on the tower frequency.